the radical rational…

in search of innovative ideas with a well-balanced approach for the math classroom

Trash the Lesson

Literally. We trashed a lesson today.  It was a lesson I’ve used for a long, long time.  Triangle Midsegment Investigation.  In recent years, I’ve used geogebra to test/ explore student findings.

It lead a small group of students through realizing midsegment is parallel and 1/2 the length of the corresponding base.


For whatever reason, it was not sinking in. 

I was getting frustrated.  On the edge of sarcasm.  I remembered a comment from my colleague last Friday, to remember to have fun with them.

The lesson wasn’t clicking with them. I wanted to trash it.

So, I picked up a student’s paper, wadded it up and instructed the entire class to follow my lead.

Noone was allowed to leave class until they threw a paper wad at someone.

They laughed. I LAUGHED.  It was totally worth it.

Then I grabbed some poster graph paper and they opened their INBs…we got the big ideas and went on to use them in a few problems…successfully.

A couple of times students made comments how all we had been doing was coming together in the new problems. They were seeing the connections.

Yep. Sometimes we just need to trash the lesson and move on.

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Painting a Bridge

In my Algebra I, we are looking at parent functions. Students said this week was quite easy, they felt they were doing 3rd grade work.  But I assured them
recognizing the parent equation and making connections to the parent graphs may seem easy, but it’s a lead-in to more intense math!

We’ve done several data collections throughout the semester, mostly linear, a few quadratic and exponential.   But today we took a look at rational with Painting the Bridge, which is embedded in a MARS lesson. 

Students are asked to sketch the relationship x:# workers and y: # hours each works to complete the given job.






Those are a good overview of what we saw.  I allowed students to ask questions about things they wondered about others’ graphs.  At first glance, a couple of the graphs may look odd, but given the chance to share thwir thimking, student reasoning made perfect sense in the real world.

Though I didn’t have an actual student create this graph, I included it on the board.


I followed the suggested questioning in the MARS lesson, which led most students to some A-ha moments.  What does point Q mean? Points S? Does it make more sense for the graph be solid or dotted? Why?

As a data collection to follow up this discussion, we picked up erasers. One student held a cup in their dominant hand and picked up one eraser at a time and placed it in the cup, we timed.  Then another student helped.  Continued adding workers and it eventually became too crowded, they were dropping erasers and slowed them down.


We compared the shape of our scatter plot and decided maybe exponential or quadrant 1 of a rational (inverse) function.

The calculator power regression resulted in
y =76x^-1.  Which gave us a chance to discuss that -1 exponent.  How it meant the inverse of multiplying by x, which was to divide by x.  So we graphed y=76/x. Nice. They were seeing the connection to our Painting the bridge discussion. 

Oh wait, how many erasers were we picking up? 78. Not bad, huh?

My goal is to give them a concrete data collection for which they can access and connect back to the math.

To end the day, they asked if they could draw a graph on the board and everyone guess the parent function name.  Sure.   They were on task and engaged so I was fine with it.
They began graphing the endpoints of their graphs,  so their classmates were finishing the graph and naming the function. It was humorous. But again, they were engaged.

I love these kids.  They were my favorites today.  It’s been a tough semester at times, but I want to end these last weeks strong. I want them to leave our classroom having grown in confidence and changed their attitude toward math.  That’s my goal. 

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Systems Linear Programming

As an intro to this lesson, I shared this scenario…

You are bidding a contract for Company ABC.  The order is for 12,000 dozen of a product and needs to be completed within 3 months.

First student question, why would anyone need 12,000 dozen of anything?  They felt this amount was a ludicrous number.  (After many summers working at Fruit of the Loom, I knew this was within reason, but a nice discussion anyhow.)

Well, is it?
According to Apple Press Info, if it’s as popular as  iPhone6…no. 

First Weekend iPhone Sales Top 10 Million, Set New Record

We figured if we had equal distribution among all 50 states, this was quite doable.

Do we have the man hours to fulfill this order in 3 months?

There are…21 (bc there were 21 students in class today) workers in this particular unit…who work 8 hours per day, each of you can complete 10 products in 1 hour.  Yes, I just made these up, but that’s what we worked with.

After a few minutes, we started sharing processes, quickly a bit of an argument – why did you do it this way? Should you have….?  Others arrived at the same solution, but with varying approaches.

I could kick myself for not taking a picture of their suggestions.  Some nice verifying one another going on.  However, they were not sure what those values represented…they could get the “right” values but lost when I asked for a label.

Watching students grapple with the numbers, made me realize how far out of reality we’ve taken students math skills.  I just want to do a better job of letting them make sense of problems themselves.

We determined it would cut it close, but we could likely finish this job, maybe requiring a bit of overtime to meet the deadline.

Now, as we make an offer for the contract, what are costs west consider? This leading to an idea of our linear programming. 
Wages, materials, utilities, insurance, packaging,  shipping, etc.  One student even said, there’s a lot to consider. Me, knodding, yes.

Is this a great example intro. Nah. But I feel it’s a nice way to show students there are many options a company must consider prior to the contract, production, sale.

Now, to the hard part.  A variety of students, some with adequate graphing skills, others struggling to find the line x> 3.

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Treats for Teachers (& a November Challenge)

As the day began, members of our student GRIT team began entering classrooms, handing teachers bags of treats.


At first glance, a bag of candy.  But when you opened, you quickly realized it was more than expected.  There were slips of paper included…handwritten notes from students.


The Student GRIT team has copied and dispersed these slips to all 1st period classes, asking each student to tell who and why they were grateful.

What a gift to read short, simple messages from some of your students / former students.

I was so grateful because it’s been a tough year.  Thank you, students for taking the time to do this for us.

They were genuine…and a treasure that has made me realize…I need to tell others thank you.  So, during the month of November, I challenge myself to write one note each day to someone to let them know they’ve made a difference to me.


Battleships & Mines Systems Review


I used Battleships and Mines as a review for systems.  We tested our mine coordinates today in the front lobby.

And then enjoyed the end of the week with the highlighters and blacklights.


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So, my dad is building a storage building.  This is the design he is using with end dimensions for his project.


He asks, “Can your students help me figure out the pitch of the roof? How long should I cut the rafters?  What angles do I use?”

Truth is, my dad likely already has them cut, ready to put together, but he knows there’s a bit of math involved that I might enjoy sharing with my class.

As I read the descritpion, I see this style is a regular octagon. 360° / 8 = 45°.  So the interior angle will be 135°.

When you cut your wood, do I split that angle in half?

I shared the diagram with other geometry teachers and Mr. H, our Carpentry Instructor at LC Area Technology Center.   I wanted to use the right approach/vocabulary Mr. H uses in his courses.  He replied almost immediately. 
The plan is to visit his classroom/shop area soon.

Now, how can I make this real for my students?  Thinking if I give them enough supplies, aka strips of construction paper to model planks of wood, and allow them to create an accurate model, describing processes for finding rafter lengths and angle measures.  Does that make sense?

An opportunity for some use of Trig, or at least a reason to use trig outside the classroom.

The website dad shared has different designs of rafters…which means we have a bank of problems to pull from.

A comment from Mr. H in our back and forth emails…

…one of the hardest to teach…geometry uses what I call the “back angle” measurement (I interpret as interior angle) and carpentry we use the “smaller” angle because we make the cuts at the intersections (if that makes sense)


Mr. H shared some online tools and resources as well to explore.

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Modeling Systems

Sort of a rambling post. But trying to make some sense of my thinking…

I always appreciate posts from @emergentmath.  This particular post made me pause, I had just completed the MARS task, Boomerangs, he references.  We are in the midst of our systems unit.

I used Mary & Alex ‘ s suggestions with beginning systems without the algebra.  Conversations were great, students’ strength in reasoning was evident.

I plan to use Geoff’s suggestion for a matching/sorting activity this werk for students to see the benefits of each type of tool to solve systems.

But where I struggle is with this standard:


I am experiencing some pushback from a handful of students who are able to reason and solve a system without actually modeling it algebraically.

Their reasoning is correct.  They verify their solutions and interpret them correctly.  They can sketch a graph yet “refuse” to model as a system of equations.  I struggle because “their math” is right on.  I realize places where algebraic models can help but I honestly can’t tell them my way is better…yet the standard says…

It feels almost like I am punishing them if I make them model it algebraically.

Then I have others who are not sure where to start.  The equations model provides them a tool, yet they will not embrace it.

How do others handle this situation in your classrooms?

I use graphical, alongside a numerical table of values, with solving/verifying with the equations, letting them see their own connections eventually.

My biggest goal for systems is to provide enough modeling for students to actually “see a context” to connect/make sense of a naked system of equations.

This is where I believe skill/drill has ruined the power and beauty of math.  Finding an intersection point but what in the world does in mean?  It’s a point on a graph. Whoopee.  Why isn’t it all taught in context as a model?

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A Trip to the Capitol

Thursday was gorgeous.  A road trip north to our state capitol, Frankfort, with my husband and my superintendent.

Ashland Inc. Teacher Achievement Awards…and I was 1 of 24!!! 


I debated on whether or not to post, but I am proud of my work in education and why not celebrate it! 
I teared up multiple times listening to the speakers.  40,000 educators in our state, over 500 nominations.  Overwhelmed to make the 24!
I am  grateful for the parent, student or colleague who took the time to nominate me. Wow. That, in itself, is such a huge honor!


Yes, of course I was a bit disappointed I wasn’t named the Teacher of the Year, but listening to their professional biographies – I was in awe just to be recognized with them. Amazing teachers.  Easy to talk to, down to earth, simply loving what they do.

Last year’s TOY, Holly Bloodworth, shared how taking guitar lessons this past year…being a learner again, made her a better teacher.  She reminded us, sometimes with standards and testing, we forget…to keep the joy in our classrooms. 

Thank you to Ashland Inc. and Kentucky Department of Education for an amazing day being honored for doing something I love.

And a shout out to this guy…for always supporting and encouraging me.  And keeping me leveled out when I get overwhelmed.


Congratulations to all Awardees!  Thank you for your passion for education!



Rethinking Assessments

After my first assessment, I cut and pasted 8 problems that were missed most or even left blank and copied.  The following day, students worked in groups to complete those problems.

The conversations were great.  I attempted to ask questions of each group member to ensure they understood their peers thinking. 

As a whole class, we discussed why several had left certain problems blank. No attempt. At.All. 
It looked hard.
Too many words.
I was confused.
It didn’t make sense.

But after working on them in small groups, several shared how most were really very doable.  or at least the way ____ explained their thinking, ot made sense.  And how likely they will be to at least give effort on those “hard” problems next to me.

My issue was, when I asked them to complete a wrong answer analysis, only about half put effort into it. Some turned in nearly blank pages. Some turned in nothing at all.  An opportunity to learn/redo and nothing.  Yet when progress reports came out, they were upset to see they had only mastered 20% of the concepts.

They were upset that I wouldn’t give them points for just redoing and I was unfair because I didn’t let the entire class redo during class time.  They wanted me to assess their work (again) they had not even practiced/studied.


My next attempt, was a group quiz – sort of modified from convo with @druinok and a post from @fnoschese and this article.   A couple of days prior to unit assessment, I gave them an overview of problems from the learni ng targets for the unit, it was an opportunity to discuss, look in their INBs, even asked questions of me.  It was quick to grade using this method Agree / disagree post-its

Somewhat better results.  But still, pushback on the redo’s.  A student was mad and asked how I expected them to remember what they had done /didn’t do on a test from the previous week.  Yes, given on a Wednesday, absent make-ups the following day, an assembly, the weekend and I passed them back on Monday. 

Round 3: Given on Friday. Stayed after school and marked each one.  Compiling My Favorite No on 4 Qs for whole – class discussion.  A mix of Qs missed most often for a quick group quiz, using the agree/disagree post-its again.  This will address majority of analysis I need from most of them, then assign remainder as HW. Final version of unit assessment following day.

I am interested to see what happens.


More Efficient Packaging

So, we ended class with a dilemma the other day.  You know the one…Dan Meyer’s 3 ACT Popcorn Picker.

A great way to review some MS concepts.  And a little popcorn treat to boot.

This week, they’ll be assigned to redesign a container…


…to a more efficient container.  Same volume, smaller surface area…


…and defend their new container…


…and write a letter (for our writing program review) to the VP of Packaging explaining how their new design can benefit the company.

Yes, it’s been around for years.  But again, a nice little review of things from a couple of years ago.


And a few Dandy Candies Gift Wrapping to see their thinking in class.

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